Given an input \(\mathbf {x}\)  such that \(\rho (\mathbf {x}) = c\) , a minimum-cardinality explanation [1]} of \(\mathbf {x}\) 
is an instance \(\mathbf {x}^{\prime }\)  such that \(\rho (\mathbf {x}^{\prime }) = c\) ,
\(\mathbf {x}^{\prime }\)  coheres with \(\mathbf {x}\)  on the ones in the sense that for any \(k \in \lbrace 1, \ldots , n\rbrace \) , if \(x^{\prime }_k = 1\)  then
\(x_k = 1\) , and \(\mathbf {x}^{\prime }\)  has a minimal number of coordinates set to 1. Roughly speaking, the features that are set to 1 in \(\mathbf {x}^{\prime }\)  are enough to explain why
\(\mathbf {x}\)  has been classified by \(\rho \)  as a positive (or as a negative) instance. Formally:
